Linear approximation methods and the best approximations of the Poisson integrals of functions from the classes $H_{ω_p}$ in the metrics of the spaces $L_p$

Abstract

We obtain upper estimates for the least upper bounds of approximations of the classes of Poisson integrals of functions from $H_{ω_p}$ for $1 ≤ p < ∞$ by a certain linear method $U_n^{*}$ in the metric of the space $L_p$. It is shown that the obtained estimates are asymptotically exact for $р = 1$: In addition, we determine the asymptotic equalities for the best approximations of the classes of Poisson integrals of functions from $H_{ω_1}$ in the metric of the space $L_1$ and show that, for these classes, the method $U_n^{*}$ is the best polynomial approximation method in a sense of strong asymptotic behavior.